Analytic Methods for Coagulation-Fragmentation Models

Growth–fragmentation–coagulation equations with unbounded coagulation kernels

“…If we use the symbol ∫0+ to denote an integral in some right neighbourhood of 0, then we may have either ∫0+dxrfalse(xfalse)=+normal∞ or ∫0+dxrfalse(xfalse)<+normal∞. When (2.8) is satisfied, the characteristics associated with the transport equation do not reach x = 0 and therefore the problem does not require a boundary condition to be specified. This case has been thoroughly researched in [1,17], and, as in [1,17], we define T 0, m by T0,mf:=Tf;1emDfalse(T0,mfalse):={f∈X0,m:rf∈AC(double-struckR+)and ddx(rf),qf∈X0,m}, where …”

“…Then D ( T 0, m ) is given by Dfalse(T0,mfalse):={f∈X0,m:rf∈AC(double-struckR+), ddx(rf),qf∈X0,m and r(x)f(x)→0 as x→0+}. To make the Hille–Yosida theorem applicable, we must determine the resolvent operator, R (λ, T 0, m ). Following [1, Section 5.2], we begin by solving λffalse(xfalse)+ddxfalse(rfalse(xfalse)ffalse(xfalse)false)+qfalse(xfalse)ffalse(xfalse)=gfalse(xfalse),1emx∈double-struckR+, where g ∈ X 0, m . On introducing antiderivatives of 1/ r and q / r , respectively, defined on double-struckR+ by Rfalse(xfalse):=∫1x1rfalse(sfalse)…”

“…Coagulation and fragmentation play a fundamental role in a number of diverse phenomena arising both in natural science and in industrial processes. Specific examples can be found in ecology, human biology, polymer and aerosol sciences, astrophysics and the powder production industry; see [1] for further details and references. A feature shared by these examples is that each involves an identifiable population of inanimate or animate objects that are capable of forming larger or smaller objects through, respectively, coalescence or break-up.…”

“…In the case of deterministic models, with either discrete or continuous size, two main approaches have been used extensively in their analysis, with one involving weak compactness arguments and the other using the theory of operator semigroups. Comprehensive treatments of each are given in [1], and there is also an excellent account in [7, ch. 36] of the semigroup approach to the discrete fragmentation equation.…”

“…The reverse process of mass gain can also occur due to the precipitation of matter from the environment. Continuous coagulation and fragmentation processes, combined with a mass transport term that leads to either mass loss or mass gain, have also been studied using functional analytic and, in particular, semigroup methods; for example, see [17–19] and [1, Section 5.2], or [20–22] where, however, the focus is on the long-term behaviour of the linear growth–fragmentation processes. The discrete versions of such models have been comprehensively analysed in [16,23].…”